Fitted finite element methods for singularly perturbed elliptic problems of convection-diffusion type

Abstract

Fitted finite element methods are constructed for a singularly perturbed convection-diffusion problem in two space dimensions. Exponential splines as basis functions are combined with Shishkin meshes to obtain a stable parameter-uniform numerical method. These schemes satisfy a discrete maximum principle. If the diffusivity parameter is bounded below by some fixed positive constant, the numerical approximations converge, in L∞, at a rate of second order. Moreover, the numerical approximations converge at a rate of first order for all values of the singular perturbation parameter.